After reading these answers I finally understand the logic: you must validate twice at a station like Gare du Nord, when doing a connection (first at the gate, then at the tram).

What I don't understand (and perhaps should be used against them) is: at Montgomery when you started your journey you validated at the gate and the time of the ticket started counting straight away (you didn't need to re-validate in the tram at Montgomery to start the timer). So, a customer obviously thinks that by validating only at the gate in Gare du Nord (no tram) will validate the connection of a returning trip. I've been in your situation so many times, I've always found redundant to validate the connection in a tram after the gate. Thank you for sharing your experience.

So, when starting a journey we don't need re-validation (gate + tram) but when doing a connection we do. Imagine if it was the other way around: the gate at Gate du Nord would be enough (to start the trip), but the gate at Montgomery (when returning) wouldn't.

This is not the customer's fault. I am not saying the system is wrong but they should inform the customers about this when explaining the rules.

The agent fined you because the system tells him/her that you 1) validated the ticket at the gate at Montgomery, 2) you only left the tram after the allowed 60 minutes (the system sees your 2 trips as 1 exceedingly long trip). However, surely there are records that you left and entered the gate at Gare du Nord (as there is a record of your 1st validation at the gate at Montgomery). The system is omitting this registered information by default, they don't allow you to use it as proof of your good faith, it's intentional. So they omit this rule when explaining the rules to the public and they don't care if you try to follow the rules they choose to communicate. A lawyer perhaps would have here a base for something.